Superconductors for power applications: an executable and web application to learn about resistive fault current limiters

The heat equation was solved on a 1-D domain and it was coupled with an electric circuit model, simulating the current sharing between the layers of the tape. A 1-D thermal assumption entails that the electrical and thermal properties of the tapes are uniform along the width and length of the tape (temperature distribution across the thickness of the tape). In figure 4, we represented the equivalent electrical model of the SFCL. The equivalent model of the SFCL is a set of resistors in parallel, representing the various layers of the tape (silver stabilizer, Hastelloy and REBCO). We do not take into account the buffer layer and the interface resistance between REBCO and silver layer. Each resistor (layer) R_i (I_i , T_i ) is defined by the electrical resistivity ρ_el,i (I_i , T_i ), the length of the tape L_tape, the thickness of the single layer h_i and the width of the tape w_tape. The subscript i identifies the layer of the tape (i.e. silver, REBCO, etc). The heat equation is coupled with the equivalent circuit model of the SFCL through the heat source. In each 1-D domain, schematically represented in figure 4, the heat equation is solved as follows:

$$\begin{align}& {\rho }_{\text{mass,}i}\left({T}_{i}\right){C}_{\mathrm{p},i}\left({T}_{i}\right)\frac{\partial {T}_{i}}{\partial t}+\frac{\partial }{\partial x}\left(-{k}_{i}\left({T}_{i}\right)\frac{\partial {T}_{i}}{\partial x}\right)={\left.\frac{{R}_{i}\left({I}_{i},{T}_{i}\right)\cdot {I}_{i}^{2}}{{{\Omega}}_{i}}-{h}_{{\text{LN}}_{2}}\cdot \left({T}_{i}-{T}_{\mathrm{0}}\right)\right\vert }_{\partial {\Omega}}.\end{align} \tag{ 2 }$$

On the left side of the one-dimensional heat equation we have the mass density ρ_mass,i (T_i ), the specific heat capacity C_p,i (T_i ) and the thermal conductivity k_i (T_i ). On the right side of equation, there are the heat source and the cooling terms. The cooling term accounts for the heat exchange with the liquid nitrogen. This is done by applying a boundary condition only on the top and the bottom layers of the tape (silver surfaces), indicated by ∂Ω. In equation (2), the heat transfer coefficient h_LN2(T_i − T₀) is a function of the temperature [20]. For better readability, the temperature dependence of h_LN2(T_i − T₀) is omitted and the transfer coefficient is simply written as h_LN2. The temperature dependence of all the thermal and electrical material properties is taken into account. Finally, the heat source comes from the joule heating effect $P={R}_{i}\cdot {I}_{i}^{2}/{{\Omega}}_{i}$, where the volume of each layer is noted as Ω_i = w_tape ⋅ L_tape ⋅ h_i .

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For modeling the REBCO resistivity, we consider the widely used power-law model with a temperature-dependent critical current I_c(T). The power-law model reads as follows:

$$\begin{equation}{\rho }_{\text{PLW}}^{\text{SC}}\left(I,T\right)=\frac{{\Sigma}\cdot {E}_{\mathrm{c}}}{{I}_{\mathrm{c}}\left(T\right)}{\left(\frac{\vert I\vert }{{I}_{\mathrm{c}}\left(T\right)}\right)}^{n-1},\end{equation} \tag{ 3 }$$

where Σ is the cross section of the REBCO layer, E_c = 1 μV cm⁻¹ is the electric field criterion and n is the constant power-law exponent, and I_c(T) reads as follows:

$$\begin{equation}{I}_{\mathrm{c}}\left(T\right)={I}_{\mathrm{c},77\enspace \text{K}}\cdot \frac{{T}_{\mathrm{c}}-T}{{T}_{\mathrm{c}}-77\enspace \text{K}}.\end{equation} \tag{ 4 }$$

In order to obtain a total resistivity that better reproduces the electrical behavior of REBCO over a wide current and temperature range, the normal state resistivity ρ_NS(T) of REBCO is added in parallel to the power-law [21]

$$\begin{equation}{\rho }_{\text{PLW}}=\frac{{\rho }_{\text{PLW}}^{\text{SC}}\left(I,T\right)\cdot {\rho }_{\text{NS}}\left(T\right)}{{\rho }_{\text{PLW}}^{\text{SC}}\left(I,T\right)+{\rho }_{\text{NS}}\left(T\right)}.\end{equation} \tag{ 5 }$$

Finally, the temperature dependence of the normal-state resistivity of REBCO is modeled with a linear relationship [22], i.e.:

$$\begin{equation*}{\rho }_{\text{NS}}\left(T\right)={\rho }_{{T}_{\mathrm{c}}}+\alpha \cdot \left(T-{T}_{\mathrm{c}}\right),\end{equation*}$$

where ${\rho }_{{T}_{\mathrm{c}}}=100\enspace \mu {\Omega}\enspace \text{cm}$ and α = 0.47 μΩ cm K⁻¹ [23].

The complete circuit model used to simulate AC fault current limitation is presented in figure 4. A sinusoidal voltage signal is imposed on the circuit, while a load resistor R_Load draws the nominal current from the source. A switch in parallel to the load resistor, when closed, simulates the fault occurring at a given time and draws the fault current through a resistor R_fault. In the simulations, V_peak can be changed and f = 50 Hz is fixed. The switch operates at t = 20 ms and the short-circuit is cleared after two periods of the sinusoidal voltage source, i.e. t = 60 ms. Finally, the prospective current is calculated as I_fault = V_peak/R_fault.

Another interesting aspect is the temperature cooling profile of the tape. Once the superconducting tape has recovered from the fault, the nominal operation of the grid must be re-established. This means that the SFCL must also be invisible to the grid (negligible impedance) and thus in the superconducting state. This condition implies that the temperature of the REBCO layer must be below its critical temperature T_c, which is around 92 K [24].

The complex non-linear heat transfer coefficient of the liquid nitrogen does not allow to estimate the cooling-temperature profile of the SFCL analytically, and thus, its recovery time. As a consequence, these quantities are estimated with numerical methods.

The time-scale at which the fault occurs (tens of ms) is very different from that of cooling, which is typically around hundreds of ms or s. The numerical solution of phenomena occurring at very different time-scales can lead to long computational times. This is undesired, especially when the user expects the simulation to run in a few minutes, as in the case of a student during a lecture. To address this issue, one can note that during the cooling of the tape the only physics involved is the heat transfer between layers and liquid nitrogen. As a consequence, we can neglect the circuital part and study the temperature profile of the tape with a second model. The second model uses the temperature profile of the tape at the end of the first simulation as initial condition, and the initial time of the second model is the final one of the first modelt₀ = t_fin. As a consequence we have:

$$\begin{equation}T{\left({t}_{0}\right)}^{2\text{nd}-\text{model}}\left(x\right)=T{\left({t}_{\text{fin}}\right)}^{1\text{st}-\text{model}}\left(x\right).\end{equation} \tag{ 6 }$$

The second model solves the heat equation on the same domains, but for a longer time (i.e. 2 s) and with a larger time step. This ensures the simulation of a time sufficiently long to observe the temperature cooling profile while having a short computational time.

The computing time of the first model (fault over 100 ms) ranges from 30 s to 40 s per run, while the computing time of the second model (tape cooling from 100 ms to 2 s) ranges from 6 s to 10 s per run. Once the simulation is run, the solver is concatenated and the overall simulation takes from 35 s to 60 s run. These computing times were obtained by running the executable on an Intel(R) Core(TM) i5-7Y57 CPU @ 1.30 GHz).

Since the main design criteria are to avoid burning the tape, we need to know whether this happened after the fault. At the end of the simulation, a dialogue box informs about the status of the tape. Specifically, the box informs if the tape reached more than 450 K and if, after 2 s ⁷ , the tape recovered, and its temperature is below 92 K. The maximum temperature displayed in the box is obtained by calculating the average temperature profile on the 1-D REBCO domain and selecting the maximum value of this curve.

In figure 5 we present the user interface. In the upper part (black-dashed box), we have the dashboard where we can find the functional buttons of the executable. Besides running the simulations ('compute') and plotting the results ('plot'), it is possible to consult a presentation describing some relevant test cases ('slide/paper') and also to export the results in form of report (.html/.doc). The parameters that can be changed and some figures of merit are indicated in figure 5.

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In the circuit drop-down list, the parameters that can be varied are V_pk, I_nom and Ratio = R_Load/R_fault. By selecting V_pk, one fixes the applied electric field E_app, since both the length of the tape (l_tape = 200 m), as well as its width (w_tape = 12 mm), are fixed by design. By choosing I_nom, one selects the nominal current flowing in the circuit in normal conditions. The critical current of the tape I_c (at 77 K and self-field conditions) is fixed accordingly, and arbitrarily, as I_c = 1.2I_nom. Finally, by selecting the ratio between the nominal and fault impedance, one selects the prospective current as I_prosp = Ratio ⋅ I_nom. At this point the user should select the tape parameters in the tape drop-down list. The parameters that can be changed, in this case, are the n-value, the silver thickness h_Ag and the Hastelloy thickness h_Hast.

Once the parameters are set and the simulation is finished, the user can consult some figures of merit such as the maximum temperature of the REBCO, the maximum limited current, and the prospective current (yellow-dashed box). Finally, the user can browse through the plots that report the quantities calculated from the model, such as the limiting current—temperature of the REBCO, the limiting current—prospective current, and the temperature cooling profile over 2 s (green-dashed box).

The first case investigates the default parameter setting, namely:

• V_pk = 12 kV (E_app = 60 V m⁻¹)
• I_nom = 500 A (I_c = 600 A)
• Ratio = 4 (I_prosp = 2000 A)
• n₀ = 30
• h_Ag = 2 μm
• h_Hast = 50 μm.

As shown in figures 6(a)–(c), with these parameters the tape is safe: the temperature stays below 450 K for the entire duration of the fault, and the limited current (<1000 A) is smaller than the prospective one (2000 A).

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In the second case all the parameters are kept as in the first case, except for the n-value. In this case the n-value is very small (i.e. n = 5) with respect to the commonly n-values of REBCO tapes (i.e. n ≈ 30 [25]). The superconductor has a smoother transition to the normal state and its behavior is closer to that of an ohmic material (n = 1).

The results (figures 6(d)–(f)) show that, for n = 5, the current flowing in the system is overlapped to the prospective current, and thus, the current is not limited (figure 6(d)). The simulated temperature profile in the REBCO predicts a very small temperature increase (ΔT ≈ 0.4 K). This means that the REBCO tape is still safe; however it is possible that other parts of the system, sensitive to overcurrents, are permanently damaged. This case highlights the crucial role played by the high non-linearity of the superconductor in making the current limitation effective.

The third case investigates a more realistic situation, where the prospective fault current can reach 10–12 times the critical current, and the applied electric field is higher. All the parameters are kept as in the first case except for V_pk = 40 kV (E_app = 200 V m⁻¹) and Ratio = 10 (I_prosp = 5000 A). The results in figures 7(a)–(c) show that the limited current (<1000 A) is smaller than the prospective one (5000 A). However, the REBCO tape is burnt since the temperature goes above 450 K. A possible solution is to add mass to the tape. This can be done by increasing the thickness of the Hastelloy layer.

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In the last case scenario we keep the same circuit parameters of the previous case, but we increase the thickness of the Hastelloy layer to 100 μm. The results are presented in figures 7(d)–(f). The electrical quantities (voltage and current) are the same, while the simulated temperature in the REBCO is substantially decreased and is below 450 K.